Electric Force is Gradient of Electric Potential Field
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Theorem
Let $R$ be a region of space in which there exists an electric potential field $F$.
The electrostatic force experienced within $R$ is the negative of the gradient of $F$:
- $\mathbf V = -\grad F$
Proof
The electrostatic force at a point of $R$ is in the direction of the greatest rate of decrease of electric potential.
That is, it is normal to the equipotential surfaces.
It also has a magnitude equal to the rate of decrease.
Hence the result.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $2$. The Gradient of a Scalar Field
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): field: 2. (field of force, force field)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): field: 2. (field of force, force field)